Aww this is so cute! Kotetsu's attachment to Koby is just adorable and I love how Barnaby just hates that. Kotetsu and Barnaby seem like such a cute couple, like sweet and spicy in a way. Kotetsu is the sweet, caring one while Barnaby is the spicy, jealous one.

Just a few grammatical errors:
"Small, pathetic squeaks escaped its small throat, whether out of fear from not know where it was since its eyes had closed shut from mucky crust over its eyes or devastating hunger neither of them had been sure." should be: Small, pathetic squeaks escaped its small throat, whether out of fear from not knowing where it was since its eyes had closed shut from mucky crust over its eyes or devastating hunger, neither of them had been sure.
"Don't worry, I washed it out." He replied, to which Barnaby shifted his glasses. should be: "Don't worry, I washed it out," he replied, to which Barnaby shifted his glasses."Don't worry, I washed it out." He replied, to which Barnaby shifted his glasses.
"But instead, Kotetsu had gently turned his new friend, giving it soft shushes and coos along with 'its ok's under his breath" should be: But instead, Kotetsu had gently turned his new friend, giving it soft shushes and coos along with 'it's ok's' under his breath
Also, towards the end, you starting switching up your tenses, which got a bit distracting. The story was being told in past tense, but you suddenly switched it to present tense. I would suggest reading the last 2 parts over again and changing all the tenses back to past.
I know there were more grammatical errors but I can't find them anymore. I think you should either read this over or get someone else to read it over for you!

This was done perfectly. Everyone was in character and it was done in such a way I could see it actually happening in the show. Kotetsu's reasoning behind Koby's name was also just so adorably sweet and I liked the ending.